2,395 research outputs found
Learning Discriminative Stein Kernel for SPD Matrices and Its Applications
Stein kernel has recently shown promising performance on classifying images
represented by symmetric positive definite (SPD) matrices. It evaluates the
similarity between two SPD matrices through their eigenvalues. In this paper,
we argue that directly using the original eigenvalues may be problematic
because: i) Eigenvalue estimation becomes biased when the number of samples is
inadequate, which may lead to unreliable kernel evaluation; ii) More
importantly, eigenvalues only reflect the property of an individual SPD matrix.
They are not necessarily optimal for computing Stein kernel when the goal is to
discriminate different sets of SPD matrices. To address the two issues in one
shot, we propose a discriminative Stein kernel, in which an extra parameter
vector is defined to adjust the eigenvalues of the input SPD matrices. The
optimal parameter values are sought by optimizing a proxy of classification
performance. To show the generality of the proposed method, three different
kernel learning criteria that are commonly used in the literature are employed
respectively as a proxy. A comprehensive experimental study is conducted on a
variety of image classification tasks to compare our proposed discriminative
Stein kernel with the original Stein kernel and other commonly used methods for
evaluating the similarity between SPD matrices. The experimental results
demonstrate that, the discriminative Stein kernel can attain greater
discrimination and better align with classification tasks by altering the
eigenvalues. This makes it produce higher classification performance than the
original Stein kernel and other commonly used methods.Comment: 13 page
Intrinsic Gaussian processes on complex constrained domains
We propose a class of intrinsic Gaussian processes (in-GPs) for
interpolation, regression and classification on manifolds with a primary focus
on complex constrained domains or irregular shaped spaces arising as subsets or
submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate
spatial domains arising as complex subsets of Euclidean space. in-GPs respect
the potentially complex boundary or interior conditions as well as the
intrinsic geometry of the spaces. The key novelty of the proposed approach is
to utilise the relationship between heat kernels and the transition density of
Brownian motion on manifolds for constructing and approximating valid and
computationally feasible covariance kernels. This enables in-GPs to be
practically applied in great generality, while existing approaches for
smoothing on constrained domains are limited to simple special cases. The broad
utilities of the in-GP approach is illustrated through simulation studies and
data examples
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